On the difference between the spectral radius and the maximum degree of graphs

Abstract

Let G be a graph with the eigenvalues λ₁(G)≥⋯≥λn(G). The largest eigenvalue of G, λ₁(G), is called the spectral radius of G. Let β(G)=Δ(G)−λ₁(G), where Δ(G) is the maximum degree of vertices of G. It is known that if G is a connected graph, then β(G)≥0 and the equality holds if and only if G is regular. In this paper we study the maximum value and the minimum value of β(G) among all non-regular connected graphs. In particular we show that for every tree T with n≥3 vertices, n−1−√(n−1) ≥ β(T) ≥ 4sin²(π/(2n+2)). Moreover, we prove that in the right side the equality holds if and only if T≅Pn and in the other side the equality holds if and only if T≅Sn, where Pn and Sn are the path and the star on n vertices, respectively.